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For a smooth 3-manifold $M$, is the natural map from the space of diffeomorphisms of $M$ to the space of homeomorphisms of $M$,

$${\sf Diff}(M) \longrightarrow {\sf Top}(M),$$

a weak homotopy equivalence? Equivalently, is the space of smooth structures on a topological 3-manifold contractible? (This is as opposed to just connected, which is the usual statement of Moise's theorem.)

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What definition of "space of smooth structures" are you using? Shouldn't it just be a single point? –  Richard Kent Dec 29 '11 at 19:16
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Here's one definition: let ${\sf Mfld}_3^{\sf sm}$ be the topological groupoid of smooth 3-manifolds with diffeomorphisms, and let ${\sf Mfld}_3$ be the topological one. Let $B$ be the classifying space functor. Then there is a natural map of spaces $$B{\sf Mfld}_3^{\sf sm} \longrightarrow B{\sf Mfld}_3.$$ One definition of the space of smooth structures on $M$ is that it is the homotopy fiber of this map, over the point $\{M\}$ in the base. One reference, for example, is Weiss & Williams' paper, Automorphisms of Manifolds. (This space is naturally a homotopy type , not a homeomorphism type.) –  John Francis Dec 29 '11 at 19:24
    
(Sorry about my first answer. I got turned around.) –  Richard Kent Dec 29 '11 at 19:26
    
Thanks for the definition. –  Richard Kent Dec 29 '11 at 19:27
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1 Answer

up vote 25 down vote accepted

$\mathsf{Diff}(S^3)$ is homotopy equivalent to $O(4)$, by Hatcher's solution to the Smale Conjecture: Hatcher, Allen E. A proof of the Smale conjecture, Diff(S3)≃O(4). Ann. of Math. (2) 117 (1983), no. 3, 553–607.

Cerf proved that the Smale conjecture implies that $\mathsf{Diff}(M) \to \mathsf{Homeo}(M)$ is a weak equivalence for all $3$--manifolds $M$, in J. Cerf, Sur les difféomorphismes de la sphère de dimension trois ( $\Gamma_4$ = 0 ), Lecture Notes in Math., vol. 53, Springer-Verlag, Berlin and New York, 1968, I think. See Linearization in 3-Dimensional Topology by Hatcher.

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That's great, thanks. –  John Francis Dec 29 '11 at 19:36
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